Template problems in group theory
Let
Let $G$ and $H$ be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism $\phi:G\to H$, then $G$ is abelian.
Let
Let $n$ be a number between $0$ and $10$. Compute $n^{111}\pmod{11}$, expressing your answer as a number between $0$ and $10$. Give as detailed a proof as you can, justifying every step, no matter who trivial you think it is.
Let
- Is the element
even or odd? Indicate your reasoning. - Find the order of
. Show all work. - Write
in disjoint cycle form. Show all work.
Let $S_n$ denote the symmetric group on $n$ letters.
\begin{enumerate}[label=\alph*)]
\item Is the element $(1\,2\,3\,4)(2\,5\,3\,4\,6)(1\,5\,3\,2\,4\,7)\in S_7$ even or odd? Indicate your reasoning.
\item Find the order of $(1\,3\,4)(2\,4\,3)(1\,3\,4)\in S_4$. Show all work.
\item Write $(1\,5\,2\,3)(2\,1\,3\,4)(1\,5\,2\,3)^{-1}\in S_5$ in disjoint cycle form. Show all work.
\end{enumerate}
Determine with proof the automorphism group
Determine with proof the automorphism group $\operatorname{Aut}(V)$ of the Klein 4-group $V=\{e,a,b,ab\}$. To what familiar group is it isomorphic?
Determine the number of group homomorphisms
Determine the number of group homomorphisms $\phi$ between the given groups. Here $K_4$ denotes the Klein four-group (also known as ${\bf Z}/2{\bf Z}\times {\bf Z}/2{\bf Z}$) and $S_3$ denotes the symmetric group on three elements.
\medskip
\begin{enumerate}[label=(\alph*)]
\item $\phi:K_4\to {\bf Z}/2{\bf Z}$
\item $\phi:{\bf Z}/2{\bf Z}\to K_4$
\item $\phi:S_3\to K_4$
\item $\phi:K_4\to S_3$
\end{enumerate}
Without using Cauchy's Theorem or the Sylow theorems, prove that every group of order 21 contains an element of order three.
Without using Cauchy's Theorem or the Sylow theorems, prove that every group of order 21 contains an element of order three.
Explicitly list all group homomorphisms
Explicitly list all group homomorphisms $f: {\bf Z}/6{\bf Z} \to {\bf Z}/12{\bf Z}$.
Let
- Describe
and in familiar terms, as groups you would study in a first algebra course. Prove your result. (Hint: Where do generators go?) - Write
down explicitly, giving its generic name and computing the order of every element. Show all work.
Let $C$ be a (possibly infinite) cyclic group, and let $\operatorname{Aut}(C)$ and $\operatorname{Inn}(C)$ be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism $\gamma$ is {\bfseries inner} if it is given by conjugation: $\gamma(b)=aba^{-1}$ for some $a\in C$.)
\begin{enumerate}[label=\alph*)]
\item Describe $\operatorname{Aut}(C)$ and $\operatorname{Inn(C)}$ in familiar terms, as groups you would study in a first algebra course. Prove your result. ({\itshape Hint:} Where do generators go?)
\item Write $\operatorname{Aut}({\bf Z}_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.
\end{enumerate}
Let
- Prove that the function
, taking to , is a surjective homomorphism. - Prove that
is isomorphic to .
Let $A_5$ denote the alternating group on a $5$-element set $\{1,2,3,4,5\}$. The set of automorphisms of $A_5$ form a group, denoted $\operatorname{Aut}(A_5)$. The group of {\bfseries conjugations} of $A_5$, denoted $\operatorname{Conj}(A_5)$, is the subgroup of $\operatorname{Aut}(A_5)$ consisting of automorphisms of the form $\gamma_s:=s(-)s^{-1}$ where $s\in A_5$. Explicitly, $\gamma_s(x)=sxs^{-1}$ for any $x\in A_5$.
\begin{enumerate}[label=\alph*)]
\item Prove that the function $\gamma:A_5\to \operatorname{Conj}(A_5)$, taking $s\in A_5$ to $\gamma_s$, is a surjective homomorphism.
\item Prove that $A_5$ is isomorphic to $\operatorname{Conj}(A_5)$.
\end{enumerate}
Suppose
Suppose $H$ is a group of order 15. Prove there does not exist a nontrivial group homomorphism $\phi:D_5\to H$, where $D_5$ is the dihedral group with ten elements.
Let
- Give an example of two non-conjugate elements of
that have the same order. - If
has maximal order, what is the order of ? - Does the element
that you found in part (2) lie in ? Fully justify your answer. - Determine whether the set
is a single conjugacy class in , where is the element you found in part (2).
Let $S_7$ denote the symmetric group.
\begin{enumerate}[label=\alph*)]
\item Give an example of two non-conjugate elements of $S_7$ that have the same order.
\item If $g\in S_7$ has maximal order, what is the order of $g$?
\item Does the element $g$ that you found in part (b) lie in $A_7$? Fully justify your answer.
\item Determine whether the set $\{h\in S_7\mid |h|=|g|\}$ is a single conjugacy class in $S_7$, where $g$ is the element you found in part (b).
\end{enumerate}
Let
- Prove
is a subgroup of . - Find an explicit generator for
and determine its order.
Let $G$ be the additive group ${\bf Z}_{2020}$ and let $H\subseteq G$ be the subset consisting of those elements with order dividing 20.
\begin{enumerate}[label=\alph*)]
\item Prove $H$ is a subgroup of $G$.
\item Find an explicit generator for $H$ and determine its order.
\end{enumerate}
Let
Let $G$ denote the set of invertible $2\times 2$ matrices with values in a field. Prove $G$ is a group by defining a group law, identity element, and verifying the axioms. Credit is based on completeness.
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